ISC 6450, Fall 2008

Methods in Complex Systems
The stuff that they told you and
that they didn't tell you in undergraduate statistics.

PhD in Complex Systems & Brain Science: Required Course
MA and PhD in Psychology: Can serve as a Quantitative Course

84633 M.W.. 12:30 PM - 1:50 PM
Behavioral Sciences, Building 12, Room 303, Boca Raton

Dr. Larry S. Liebovitch
telephone: 561.297.2239
http://www.ccs.fau.edu/~liebovitch/larry.html

DESCRIPTION:
Linear, parametric, non-parametric, and nonlinear analysis of experimental data. This course helps you to understand the assumptions used in these statistical methods and which statistical methods are best for analyzing different types of data. Presents the classical statistical analysis and inference of linear systems that have a small number of noninteracting pieces and how those statistical methods and analysis procedures are different for nonlinear complex systems with many pieces that interact strongly with each other, such as fractals and chaos.

TEXTBOOKS:
required
M. R. Spiegel and L. J. Stephens. Schaum's Outlines: Statistics, 4th Edition. McGraw Hill, New York, ISBN 978-0-07-148584-5
recommended
M. Hollander and D. A. Wolf. Nonparametric Statistical Methods. John Wiley and Sons, New York, 1973.
L. S. Liebovitch. Fractals and Chaos Simplified for the Life Sciences. Oxford University Press, New York, 1998.



COURSE POLICIES:

Attendance: Students are expected to attend all scheduled classes. Should it become necessary for a student to miss a class, the student is responsible for the material covered during that class. It is the responsibility of the student to withdraw from this class, should that status be desired - the instructor cannot withdraw students from the course.

Reading the Textbook: PLEASE read the chapters assigned prior to the class session in which the material will be presented.

Homework Problems: Please try to turn in the homework problems on time. NONE will be accepted after the last meeting of the class.

Grading: The grade will be determined entirely from the homework problems. There will be no exams.

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Topics

(Chapters in Spiegel textbook)


What is science? (paradigms, observation, theory, and experiments)
Why measure more than once? (mathematical basis, distributions) (Ch. 1-2,6,7)
Characterizing data (moments, central tendency, dispersion) (Ch. 3-5)
Fitting functional relationships to data (least squares, correlation, Hogdes-Lehman, linear, log) (Ch. 13-14)
Review of Homework 1: mean, variance, Gaussian distributions
Designing Experiments (controls, blind, double blind, surrogate end points)
Review of Homework 2: least squares and correlation coefficients
Statistical tests - Gaussian (z, t, F, Chi Square) (Ch. 8-12)
Statistical tests - Nonparametric (rank order, Hodges-Lehman) (Ch. 17)
Review of Homework 3: parametric statistical tests
Analysis of factorial experiment (main effects, interactions, ANOVA) (Ch. 16)
Review of Homework 4: non-parametric tests
Introduction to fractals
Review of Homework 5: ANOVA
Fractal methods of analysis (power spectra, Hurst, F(n), variance, Fano, coeff. var.)
Introduction to chaos (nonlinear dynamics)
Chaos methods of analysis (phase space, dimension, Liapunov, power spectra)
Analysis of data from student experiments

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Homework

Homework #1
1) Compute the pdf (probability density function) of the data in Spiegel problem 2.2 (page 42).
2) Compute the mean, standard deviation of the data in #1.
3) Plot a Gaussian curve, using the mean and standard deviation of #2 on the same pdf as obtained in #1.
4) Compute the 3rd and 4th moments of this data. Compare the 4th moment with 3s⁴.
5) Splatter ink on a piece of paper. Measure the diameter of 100 ink spots. Compute the pdf of the diameters.

Homework #2
1) Use the data in Spiegel, Table 13.5, Problem 13.10, Page 328. Write a computer program that uses the least squares method to determine the best straight line that represents the function of: a) Weight as a function of Height as well as b) Height as a function of Weight
2) Compute the correlation function r for the two lines in problem #1.
(OPTIONAL) 3) Write a computer program that uses the nonparametric Hodges-Lehmann estimator for the slope and intercept for the two lines in problem #1.

Homework #3
1) Use the Chi-Square test to determine if the data from the first homework (Spiegel Problem 2.2, page 42) is a Gaussian or not.
2) Spiegel Problem 10.9, page 256.
3) Spiegel Problem 10.18, page 264.
4) Spiegel Problem 11.6, page 282-283.
5) Spiegel Problem 11.8, page 284.

Homework #4
1) When a series of numbers is uncorrelated the fractal parameter called the Hurst H coefficient is approximately equal to 0.5. We found that H=0.61 in the data from one patient. If the data from the patient was correlated, then when we shuffle it, those correlations should be removed and the value of H should be closer to 0.5. To determine if the H measured from the patient is different from 0.5, we shuffled that data and then measured H. We did this 16 times. In these shuffled data sets we found that H=0.55, 0.56, 0.64, 0.54, 0.51, 0.63, 0.55, 0.48, 0.62, 0.55, 0.54, 0.51, 0.53, 0.54, 0.54, 0.50. Is the H from the patient statistically significantly higher than that of uncorrelated data? (HINT: We found that 3 out of 16 times the value of H from the patient was greater than the uncorrelated randomized data. Use a binomial test to determine if 3 out of 16 is statistically significant).
(OPTIONAL) 2) Perform a rank-sum (Wilcoxon) test on the data from Table 1. page 69 in Hollander and Wolfe to determine if there is a difference in the water permeability between the two groups.

Homework #5
1) Spiegel Problems 16.4, 16.5, and 16.6, pages 415-418.
2) Spiegel Problem 16.11, pages 422-424.
3) Spiegel Problem 16.13, pages 425-431.